Subgroups — C₂×#20241487 #26131179

Order 12032.

614 conjugacy classes of subgroups.

The subgroup lattice has 304 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×#13093960	C₂	1	1	yes	maximal, characteristic
2	C₂²×#13093956	C₂	1	1	yes	maximal, characteristic
3	#20241487	C₂	1	4	yes	maximal, direct product factor
4	C₂²×M₃₂×C₄₇	C₂	1	1	yes	maximal, characteristic
5	C₂²×#10625329	—	2	1	no	—
6	C₂²×C₄⋊Dic₄₇	—	2	1	no	—
7	C₂×#13093960	—	2	1	no	—
8	C₂×#13093960	—	2	2	no	—
9	C₂×#13093960	C₄	1	4	yes	complemented
10	C₂×#13093960	C₂²	1	2	yes	—
11	C₂×#13093956	—	2	1	no	—
12	C₂×#13093956	—	2	2	no	—
13	C₂×#13093956	C₄	1	4	yes	—
14	C₂×#13093956	C₂²	1	2	yes	—
15	C₂³×C₈×C₄₇	C₂²	1	1	yes	characteristic
16	C₂×M₃₂×C₄₇	—	2	2	no	—
17	C₂×M₃₂×C₄₇	—	2	2	no	—
18	C₂×M₃₂×C₄₇	C₄	1	2	yes	complemented
19	C₂×M₃₂×C₄₇	C₂²	1	2	yes	—
20	C₂²×C₁₆×C₄₇	C₄	1	2	yes	complemented
21	C₂³×Dic₄₇	—	4	1	no	—
22	C₂×#10625329	—	2	4	no	—
23	C₂×#10625329	—	2	2	no	—
24	C₂×#10625329	—	4	2	no	—
25	C₂×#10625329	—	4	1	no	—
26	C₂²×C₄₇⋊C₈	—	4	1	no	—
27	C₂×C₄⋊Dic₄₇	—	2	4	no	—
28	C₂×C₄⋊Dic₄₇	—	2	2	no	—
29	C₂×C₄⋊Dic₄₇	—	4	2	no	—
30	C₂×C₄⋊Dic₄₇	—	4	1	no	—
31	#13093960	—	2	4	no	—
32	#13093960	—	2	2	no	—
33	#13093960	C₂×C₄	1	4	yes	complemented
34	#13093956	—	2	4	no	—
35	#13093956	—	2	2	no	—
36	#13093956	C₂×C₄	1	4	yes	—
37	C₂³×C₄×C₄₇	D₄	1	1	yes	characteristic
38	C₂²×C₈×C₄₇	C₂×C₄	1	1	yes	characteristic
39	C₂²×C₈×C₄₇	C₂×C₄	1	1	yes	characteristic
40	C₂²×C₈×C₄₇	C₂×C₄	1	2	yes	—
41	C₂²×C₈×C₄₇	C₂×C₄	1	2	yes	—
42	C₂²×C₈×C₄₇	D₄	1	1	yes	characteristic
43	C₂²×C₈×C₄₇	D₄	1	2	yes	—
44	C₂²×C₈×C₄₇	D₄	1	2	yes	—
45	C₂²×C₈×C₄₇	Q₈	1	1	yes	characteristic
46	C₂²×C₈×C₄₇	Q₈	1	1	yes	characteristic
47	C₂²×C₈×C₄₇	C₂³	1	1	yes	characteristic
48	M₃₂×C₄₇	—	2	4	no	—
49	M₃₂×C₄₇	—	2	2	no	—
50	M₃₂×C₄₇	—	2	2	no	—
51	C₂×C₁₆×C₄₇	—	2	4	no	—
52	C₂×C₁₆×C₄₇	C₂×C₄	1	4	yes	complemented
53	C₂⁴×C₄₇	SD₁₆	1	1	yes	characteristic
54	C₂²×C₄×C₄₇	C₂²⋊C₄	1	2	yes	—
55	C₂²×C₄×C₄₇	C₂²⋊C₄	1	2	yes	—
56	C₂²×C₄×C₄₇	C₄⋊C₄	1	1	yes	characteristic
57	C₂²×C₄×C₄₇	C₄⋊C₄	1	1	yes	characteristic
58	C₂²×C₄×C₄₇	D₈	1	2	yes	—
59	C₂²×C₄×C₄₇	SD₁₆	1	1	yes	characteristic
60	C₂²×C₄×C₄₇	SD₁₆	1	1	yes	characteristic
61	C₂²×C₄×C₄₇	SD₁₆	1	1	yes	characteristic
62	C₂²×C₄×C₄₇	Q₁₆	1	2	yes	—
63	C₂²×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
64	C₂²×Dic₄₇	—	4	4	no	—
65	C₂²×Dic₄₇	—	4	2	no	—
66	C₂²×Dic₄₇	—	8	4	no	—
67	C₂×C₈×C₄₇	C₄²	1	2	yes	—
68	C₂×C₈×C₄₇	C₄²	1	2	yes	—
69	C₂×C₈×C₄₇	C₂²⋊C₄	1	2	yes	—
70	C₂×C₈×C₄₇	C₂²⋊C₄	1	2	yes	—
71	C₂×C₈×C₄₇	C₂²⋊C₄	1	2	yes	—
72	C₂×C₈×C₄₇	C₂²⋊C₄	1	2	yes	—
73	C₂×C₈×C₄₇	C₄⋊C₄	1	1	yes	characteristic
74	C₂×C₈×C₄₇	C₄⋊C₄	1	1	yes	characteristic
75	C₂×C₈×C₄₇	C₄⋊C₄	1	2	yes	—
76	C₂×C₈×C₄₇	C₄⋊C₄	1	2	yes	—
77	C₂×C₈×C₄₇	C₄⋊C₄	1	2	yes	—
78	C₂×C₈×C₄₇	C₄⋊C₄	1	2	yes	—
79	C₂×C₈×C₄₇	C₂²×C₄	1	1	yes	characteristic
80	C₂×C₈×C₄₇	C₂²×C₄	1	1	yes	characteristic
81	C₂×C₈×C₄₇	C₂²×C₄	1	1	yes	characteristic
82	C₂×C₈×C₄₇	C₂×D₄	1	1	yes	characteristic
83	C₂×C₈×C₄₇	C₂×D₄	1	1	yes	characteristic
84	C₂×C₈×C₄₇	C₂×Q₈	1	1	yes	characteristic
85	C₄⋊Dic₄₇	—	2	4	no	—
86	C₄⋊Dic₄₇	—	4	4	no	—
87	C₄⋊Dic₄₇	—	4	2	no	—
88	#10625329	—	2	4	no	—
89	#10625329	—	4	4	no	—
90	#10625329	—	4	2	no	—
91	C₂×C₄₇⋊C₈	—	4	4	no	—
92	C₂×C₄₇⋊C₈	—	4	2	no	—
93	C₁₆×C₄₇	—	2	4	no	—
94	C₂³×C₄₇	—	2	4	no	—
95	C₂³×C₄₇	#102	1	2	yes	—
96	C₂³×C₄₇	#103	1	2	yes	—
97	C₂³×C₄₇	#106	1	1	yes	characteristic
98	C₂³×C₄₇	#106	1	1	yes	characteristic
99	C₂³×C₄₇	C₂×SD₁₆	1	1	yes	characteristic
100	C₂×C₄×C₄₇	#95	1	2	yes	—
101	C₂×C₄×C₄₇	#95	1	2	yes	—
102	C₂×C₄×C₄₇	#102	1	2	yes	—
103	C₂×C₄×C₄₇	#102	1	2	yes	—
104	C₂×C₄×C₄₇	#102	1	2	yes	—
105	C₂×C₄×C₄₇	#103	1	2	yes	—
106	C₂×C₄×C₄₇	#103	1	2	yes	—
107	C₂×C₄×C₄₇	#103	1	2	yes	—
108	C₂×C₄×C₄₇	#106	1	1	yes	characteristic
109	C₂×C₄×C₄₇	#106	1	1	yes	characteristic
110	C₂×C₄×C₄₇	#107	1	2	yes	—
111	C₂×C₄×C₄₇	#107	1	2	yes	—
112	C₂×C₄×C₄₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
113	C₂×C₄×C₄₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
114	C₂×C₄×C₄₇	C₂×C₄⋊C₄	1	1	yes	characteristic
115	C₂×C₄×C₄₇	C₂×D₈	1	1	yes	characteristic
116	C₂×C₄×C₄₇	C₂×SD₁₆	1	1	yes	characteristic
117	C₂×C₄×C₄₇	C₂×Q₁₆	1	1	yes	characteristic
118	C₂×Dic₄₇	—	4	4	no	—
119	C₂×Dic₄₇	—	8	8	no	—
120	C₂×Dic₄₇	—	8	4	no	—
121	C₈×C₄₇	#95	1	2	yes	—
122	C₈×C₄₇	#95	1	2	yes	—
123	C₈×C₄₇	C₂×C₄²	1	1	yes	characteristic
124	C₈×C₄₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
125	C₈×C₄₇	C₂×C₄⋊C₄	1	1	yes	characteristic
126	C₈×C₄₇	C₂×C₄⋊C₄	1	1	yes	characteristic
127	C₄₇⋊C₈	—	4	4	no	—
128	C₂×#1383	—	47	1	no	maximal, Sylow
129	C₂²×C₄₇	—	2	4	no	—
130	C₂²×C₄₇	—	2	4	no	—
131	C₂²×C₄₇	—	2	2	no	—
132	C₂²×C₄₇	—	2	2	no	—
133	C₂²×C₄₇	#340	1	2	yes	—
134	C₂²×C₄₇	#340	1	2	yes	—
135	C₂²×C₄₇	#365	1	4	yes	—
136	C₂²×C₄₇	C₂×#102	1	1	yes	characteristic
137	C₂²×C₄₇	C₂×#103	1	1	yes	characteristic
138	C₂²×C₄₇	C₂×#106	1	1	yes	characteristic
139	C₄×C₄₇	#340	1	2	yes	—
140	C₄×C₄₇	#340	1	2	yes	—
141	C₄×C₄₇	C₂×#95	1	1	yes	characteristic
142	C₄×C₄₇	C₂×#102	1	1	yes	characteristic
143	C₄×C₄₇	C₂×#103	1	1	yes	characteristic
144	C₄×C₄₇	C₂×#107	1	1	yes	characteristic
145	Dic₄₇	—	8	4	no	—
146	C₂²×M₃₂	D₄₇	1	1	yes	characteristic
147	C₂²×#108	—	47	1	no	—
148	C₂²×#106	—	47	1	no	—
149	#1383	—	47	4	no	—
150	C₂×C₄₇	—	2	1	no	—
151	C₂×C₄₇	—	2	2	no	—
152	C₂×C₄₇	—	2	1	no	—
153	C₂×C₄₇	#1383	1	4	yes	complemented
154	C₂×C₄₇	C₂×#340	1	1	yes	characteristic
155	C₂×C₄₇	C₂×#365	1	2	yes	—
156	C₂²×M₁₆	—	94	1	no	—
157	C₂³×C₈	C₂×D₄₇	1	1	yes	characteristic
158	C₂²×C₄⋊C₄	—	94	1	no	—
159	C₂×M₃₂	—	2	2	no	—
160	C₂×M₃₂	—	2	2	no	—
161	C₂×M₃₂	Dic₄₇	1	2	yes	complemented
162	C₂×M₃₂	C₂×D₄₇	1	2	yes	—
163	C₂²×C₁₆	Dic₄₇	1	2	yes	complemented
164	C₂×#108	—	47	4	no	—
165	C₂×#108	—	47	2	no	—
166	C₂×#108	—	94	1	no	—
167	C₂×#108	—	94	2	no	—
168	C₂×#106	—	47	4	no	—
169	C₂×#106	—	47	2	no	—
170	C₂×#106	—	94	1	no	—
171	C₂×#106	—	94	2	no	—
172	C₄₇	C₂×#1383	1	1	yes	characteristic, complemented, Sylow
173	C₂³×C₄	—	188	1	no	—
174	C₂³×C₄	D₁₈₈	1	1	yes	characteristic
175	C₂×M₁₆	—	94	4	no	—
176	C₂×M₁₆	—	94	2	no	—
177	C₂×M₁₆	—	188	1	no	—
178	C₂×M₁₆	—	188	2	no	—
179	C₂²×C₈	—	188	1	no	—
180	C₂²×C₈	Dic₉₄	1	1	yes	characteristic
181	C₂²×C₈	Dic₉₄	1	1	yes	characteristic
182	C₂²×C₈	C₄×D₄₇	1	2	yes	—
183	C₂²×C₈	C₄×D₄₇	1	2	yes	—
184	C₂²×C₈	D₁₈₈	1	1	yes	characteristic
185	C₂²×C₈	C₂×Dic₄₇	1	1	yes	characteristic
186	C₂²×C₈	C₂×Dic₄₇	1	1	yes	characteristic
187	C₂²×C₈	C₂²⋊D₄₇	1	2	yes	—
188	C₂²×C₈	C₂²⋊D₄₇	1	2	yes	—
189	C₂²×C₈	C₂²×D₄₇	1	1	yes	characteristic
190	C₂×C₄⋊C₄	—	94	4	no	—
191	C₂×C₄⋊C₄	—	94	2	no	—
192	C₂×C₄⋊C₄	—	188	1	no	—
193	C₂×C₄⋊C₄	—	188	2	no	—
194	M₃₂	—	2	4	no	—
195	M₃₂	—	2	2	no	—
196	M₃₂	—	2	2	no	—
197	C₂×C₁₆	—	2	4	no	—
198	C₂×C₁₆	C₂×Dic₄₇	1	4	yes	complemented
199	#108	—	47	4	no	—
200	#108	—	94	4	no	—
201	#108	—	94	2	no	—
202	#106	—	47	4	no	—
203	#106	—	94	4	no	—
204	#106	—	94	2	no	—
205	C₂⁴	#10625325	1	1	yes	characteristic
206	C₂²×C₄	—	376	4	no	—
207	C₂²×C₄	—	188	4	no	—
208	C₂²×C₄	—	188	2	no	—
209	C₂²×C₄	#10625325	1	1	yes	characteristic
210	C₂²×C₄	#10625325	1	1	yes	characteristic
211	C₂²×C₄	#10625325	1	1	yes	characteristic
212	C₂²×C₄	D₃₇₆	1	2	yes	—
213	C₂²×C₄	Dic₁₈₈	1	2	yes	—
214	C₂²×C₄	C₄⋊Dic₄₇	1	1	yes	characteristic
215	C₂²×C₄	C₄⋊Dic₄₇	1	1	yes	characteristic
216	C₂²×C₄	(C₂×D₄₇)⋊C₄	1	2	yes	—
217	C₂²×C₄	(C₂×D₄₇)⋊C₄	1	2	yes	—
218	C₂²×C₄	C₂×D₁₈₈	1	1	yes	characteristic
219	M₁₆	—	94	4	no	—
220	M₁₆	—	188	4	no	—
221	M₁₆	—	188	2	no	—
222	C₂×C₈	—	188	4	no	—
223	C₂×C₈	—	188	2	no	—
224	C₂×C₈	C₄×Dic₄₇	1	2	yes	—
225	C₂×C₈	C₄×Dic₄₇	1	2	yes	—
226	C₂×C₈	Dic₄₇⋊C₄	1	2	yes	—
227	C₂×C₈	Dic₄₇⋊C₄	1	2	yes	—
228	C₂×C₈	Dic₄₇⋊C₄	1	2	yes	—
229	C₂×C₈	Dic₄₇⋊C₄	1	2	yes	—
230	C₂×C₈	C₄⋊Dic₄₇	1	1	yes	characteristic
231	C₂×C₈	C₄⋊Dic₄₇	1	1	yes	characteristic
232	C₂×C₈	(C₂×D₄₇)⋊C₄	1	2	yes	—
233	C₂×C₈	(C₂×D₄₇)⋊C₄	1	2	yes	—
234	C₂×C₈	C₂²⋊Dic₄₇	1	2	yes	—
235	C₂×C₈	C₂²⋊Dic₄₇	1	2	yes	—
236	C₂×C₈	C₂×Dic₉₄	1	1	yes	characteristic
237	C₂×C₈	C₂×C₄×D₄₇	1	1	yes	characteristic
238	C₂×C₈	C₂×C₄×D₄₇	1	1	yes	characteristic
239	C₂×C₈	C₂²×Dic₄₇	1	1	yes	characteristic
240	C₂×C₈	C₂×C₂²⋊D₄₇	1	1	yes	characteristic
241	C₂×C₈	C₂×C₂²⋊D₄₇	1	1	yes	characteristic
242	C₄⋊C₄	—	94	4	no	—
243	C₄⋊C₄	—	188	4	no	—
244	C₄⋊C₄	—	188	2	no	—
245	C₁₆	—	2	4	no	—
246	C₂³	—	2	4	no	—
247	C₂³	#13093946	1	2	yes	—
248	C₂³	#13093949	1	2	yes	—
249	C₂³	#13093956	1	1	yes	characteristic
250	C₂³	#13093956	1	1	yes	characteristic
251	C₂³	C₂×#10625325	1	1	yes	characteristic
252	C₂×C₄	—	376	8	no	—
253	C₂×C₄	—	376	4	no	—
254	C₂×C₄	—	188	4	no	—
255	C₂×C₄	C₄₇⋊#95	1	2	yes	—
256	C₂×C₄	C₄₇⋊#95	1	2	yes	—
257	C₂×C₄	#13093946	1	2	yes	—
258	C₂×C₄	#13093946	1	2	yes	—
259	C₂×C₄	#13093946	1	2	yes	—
260	C₂×C₄	#13093949	1	2	yes	—
261	C₂×C₄	#13093949	1	2	yes	—
262	C₂×C₄	#13093949	1	2	yes	—
263	C₂×C₄	#13093956	1	1	yes	characteristic
264	C₂×C₄	#13093956	1	1	yes	characteristic
265	C₂×C₄	#13093958	1	2	yes	—
266	C₂×C₄	#13093958	1	2	yes	—
267	C₂×C₄	C₂×(C₂×D₄₇)⋊C₄	1	1	yes	characteristic
268	C₂×C₄	C₂×(C₂×D₄₇)⋊C₄	1	1	yes	characteristic
269	C₂×C₄	C₂×C₄⋊Dic₄₇	1	1	yes	characteristic
270	C₂×C₄	C₂×D₃₇₆	1	1	yes	characteristic
271	C₂×C₄	C₂×#10625325	1	1	yes	characteristic
272	C₂×C₄	C₂×Dic₁₈₈	1	1	yes	characteristic
273	C₈	—	188	4	no	—
274	C₈	C₄₇⋊#95	1	2	yes	—
275	C₈	C₄₇⋊#95	1	2	yes	—
276	C₈	C₂×C₄×Dic₄₇	1	1	yes	characteristic
277	C₈	C₂×C₂²⋊Dic₄₇	1	1	yes	characteristic
278	C₈	C₂×Dic₄₇⋊C₄	1	1	yes	characteristic
279	C₈	C₂×Dic₄₇⋊C₄	1	1	yes	characteristic
280	C₂²	—	2	4	no	—
281	C₂²	—	2	4	no	—
282	C₂²	—	2	2	no	—
283	C₂²	—	2	2	no	—
284	C₂²	#15688886	1	2	yes	—
285	C₂²	#15688886	1	2	yes	—
286	C₂²	#15688946	1	4	yes	—
287	C₂²	C₂×#13093946	1	1	yes	characteristic
288	C₂²	C₂×#13093949	1	1	yes	characteristic
289	C₂²	C₂×#13093956	1	1	yes	characteristic
290	C₄	—	376	4	no	—
291	C₄	#15688886	1	2	yes	—
292	C₄	#15688886	1	2	yes	—
293	C₄	C₂×C₄₇⋊#95	1	1	yes	characteristic
294	C₄	C₂×#13093946	1	1	yes	characteristic
295	C₄	C₂×#13093949	1	1	yes	characteristic
296	C₄	C₂×#13093958	1	1	yes	characteristic
297	C₂	—	2	1	no	—
298	C₂	—	2	1	no	—
299	C₂	—	2	2	no	—
300	C₂	#20241487	1	4	yes	direct product factor
301	C₂	C₂×#15688886	1	1	yes	characteristic
302	C₂	C₂×#15688946	1	2	yes	—

	subgroup	G/H
center	C₂³	#13093956
commutator	C₈×C₄₇	C₂×C₄²
Frattini	C₂²×C₈	C₂²×D₄₇
Fitting	C₂²×M₃₂×C₄₇	C₂
socle	C₂³×C₄₇	#106
prime radical	C₂⁴×C₄₇	SD₁₆
cosocle	C₂²×C₈×C₄₇	C₂³
hypercenter	C₂²×M₃₂	D₄₇
hypocenter	C₄₇	C₂×#1383
LCS	G ▷ C₈×C₄₇ ▷ C₄×C₄₇ ▷ C₂×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂³ ◁ C₂³×C₄ ◁ C₂³×C₈ ◁ C₂²×M₃₂

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

N	H ≅ G/N	type
3. #20241487	C₂	direct (N × H)
9. C₂×#13093960	C₄	semidirect
18. C₂×M₃₂×C₄₇	C₄	semidirect
20. C₂²×C₁₆×C₄₇	C₄	semidirect
33. #13093960	C₂×C₄	semidirect
52. C₂×C₁₆×C₄₇	C₂×C₄	semidirect
153. C₂×C₄₇	#1383	semidirect
161. C₂×M₃₂	Dic₄₇	semidirect
163. C₂²×C₁₆	Dic₄₇	semidirect
172. C₄₇	C₂×#1383	semidirect
198. C₂×C₁₆	C₂×Dic₄₇	semidirect
300. C₂	#20241487	direct (N × H)

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroup	index
C₂²×#13093960	2
C₂²×#13093956	2
#20241487	2
C₂²×M₃₂×C₄₇	2
C₂×#1383	47

Subgroups — C2×#20241487 #26131179

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#20241487 #26131179